Subject: mathematics question
Date: Sun, 2 May 1999 01:47:43 +0100

A shape preserves its shape if a rotation, translation or scaling is performed on it. Are these the only continuous transformations which have this property? These transformations if performed on the parts and then summed have the same effect as the transformation being applied to the whole; are these linear transformations? Who, and what area of mathematics has classified all transformations of this type completely?

jmc

Hi,

The shape-preserving transformations in the plane:
 Isometries: rigid motions = displacements = direct isometries (which consist of ROTATIONS and TRANSLATIONS) opposite isometries (which consist of REFLECTIONS and GLIDE REFLECTIONS ( or simply, GLIDES); a glide consists of a translation along a line followed by reflection in the line -- if you picture this line as a straight path through snow, then consecutive footprints are related by a glide.) Similarities: direct: a DILATIVE ROTATION (a dilatation, or scaling, followed by a rotation (possibly through a zero angle) about the fixed point of the scaling) opposite: a dilative reflection (a dilatation followed by reflection in a line through the fixed point)

The shape-preserving transformations of 3-space.
 Isometries: direct: rotation (about a line), translation, and twist (= screw displacement, which is a translation along a line followed by a rotation about the line) opposite: a glide (translation along a plane followed by reflection in the plane) and a rotatory reflection = rotatory inversion (rotation about a line followed by reflection in a plane perpendicular to the line). Similarities A dilative rotation is the only possibility -- it can be direct or opposite depending on the dilatation.

REFERENCE. I have taken the terminology from H.S.M. Coxeter's INTRODUCTION TO GEOMETRY (CHAPTERS 3, 5, AND 7). There are many textbooks that deal with isometries; many high-school texts have a chapter on the topic.

HISTORY. Geometers have always thought dynamically, using transformations long before the concepts were carefully defined. For example, some of Euclid's proofs (300 BC) rely on isometries. In 1872 the German mathematician Felix Klein proposed that transformations be used as the basis for geometry, with Euclid's axioms being supplemented by transformation axioms. The modern interest in transformation groups can be traced back to him, although he would probably admit that none of his ideas were original.

Finally, YES, all isometries and similarities are linear transformations: They can be described using matrices acting on vector spaces.

Cheers
Chris

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